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## Integrated math 3

### Course: Integrated math 3>Unit 1

Lesson 2: Average rate of change of polynomials

# Finding average rate of change of polynomials

Finding the average rate of change of f(x)=x³-4x over the interval [-2,3].

## Want to join the conversation?

• Okay, this is over my head. In my school we take Algebra 1, then Geometry, then Algebra 2 (This upcoming year I’m going to take Algebra 2). So that may be why I’m so confused. How does [-2,3] affect what he did to solve the problem? And I don’t really get why he used the table to get the answer? And what he was looking for in the table? Can somebody explain or know a video that could help me understand?
• [-2,3] is fancy math notation. The [ ] means you include those values, so the interval we want to examine goes from -2 to +3. It really means: plug -2 in for x and find the y value. Then plug 3 in for x and find the y value. How much does the input (x-value) change from -2 to 3? (it changes by +5). How much does the output (y value) change over the x interval of -2 to 3? (it changes by +15). Average rate of change means change in output (change in y, or f(x)), divided by change in input (change in x). Sal used a table to show this algebraically. Using algebra means you don't have to know what the function looks like on a graph; you can still do the problem, BUT being able to see it with algebra and visually (on a graph) will allow for a deeper understanding and more connection of what the numbers mean.
• i m genuinely confused, i don't understand the topic.
• Hello! I hope this helps.

The definition of a Function is "a relationship or expression involving one or more variables" (credits to the Oxford English dictionary and Google). Non-linear functions are more confusing than linear functions because they are less concrete and, well, linear than an Algebra I function.

A Function is the relation of things through input and output. The most common name for a function is (f), but it can be anything. What goes into a function (the input) is put in parentheses, such as f(x). The X is just a placeholder for a number. If you get the function f(x) = x^2, you plug in the numbers that they give you. Here is an example of the Function listed above if you plug in the number 5:
f(5) = 5^2.

You can also make a function look like this:
y=x^2
There is no name for the Function, but there is an input (x), a relationship between the two numbers (^2), and an output (y). The equation above is a function because a function relates an input to an output.

To test if a graphed line or curve is a function, make sure that the line has no vertical parts, and having part of it go through the same x point. For example, having something go through the point (7,8), and then going through the point (7,9). A Function having two inputs is a valid curve or line, but not a function. HOWEVER, you can have two repeated y points. For example, having one going through the points (9,8) and then having them go through the points (7,8). That means that there can be multiple outputs, but only one input.

Functions are polynomials, given that they relate to one or more variables (this allows for monomials - things with only one number, like 8). Sal is talking about how to find the average rate of change (like a slope) for a non-linear function.

Hope this helps, Mimi (and anyone else with that question). Let me know if anything was unclear.
• Can somebody explain the graph that he drew? I understood everything up to that point.
• The graph Sal drew was supposed to help visualize the average rate of change (aka the slope).
• At around , Sal starts talking about the chart, but when he was multiplying, I did not understand how he got y=f(x)=-8+8=0
• f(x) = x^3 - 4x

so in other words you plug in numbers for x to find what y is since y = f(x). In this instance Sal wanted t find f(-2). Or in other words, if you graphed f(x) where is the line of the graph when x=-2

All this means is that you take f(x) and plug -2 in for all of the xs. the way this is written out is f(-2)

so f(x) = x^3 - 4x so if you plug in -2 for all xs this gets us (-2)^3 - 4x. (-2)^3 = -2*-2*-2 = 8 and -4*-2 = 8 so (-2)^3 - 4x = -8 + 8 = 0.

Let me know if that didn't help.
• why does sal make y = f(3) = 27-12=15. Where does the 15 come from. The other part I understand.
• The original equation is f(x) = x³-4x.
f(3) is simply replacing everything you see with x with 3; thus you'd get:
f(3) = 3³-4(3)
= 27 - 12
= 15
f(3) = 15, hopefully that helps!
• So on the graph, I get the change in y and the change in x. What I don't get is the loopy line, and what defines the loopy line. The most I can make of it is the start and end. But couldn't we just use a straight line? Why does it have to be so curvy, and what purpose does it serve?
• Well for this example, that loopy line is a polynomial function. More specfically, it's f(x) = x³-4x. Any type of polynomial with an degree (highest exponent) as two or more are naturally curvy. Desmos would help you lots + plugging numbers in !

The purpose of curvy lines and not straight lines, is not everything is going to be as linear as we may thing, world speaking. Mathematics does not only cover just straight lines, but curvy, fulling,weirdly interesting functions very philosophical lmao but hopefully that helps.
• if 0+15=15 so how -2+3=5 ?
• So what he was trying to say its the distance if it was on a number line then -2 is five away from 3
• Whats the difference between an interval and point? Like [-2,3] is an interval but (-2,3) is point ? I though that for a point and interval the corresponding numbers are values of x and y (x,y). But in this example he uses [-2,3] as inputs for the function meaning that they are both x values?